The electromagnetic force is the only one of the four fundamental forces of nature that is compatible with the four realms of mechanics (classical, quantum, relativistic and relativistic quantum mechanics) and therefore, the study of electrodynamics is fundamental to understanding how the laws of physics may be unified, but also to identify gaps in our knowledge. Drawing upon the foundations of classical electromagnetism and optics laid in the undergraduate physics major, this unit provides an advanced-level treatment of topics in electrodynamics and photonics underlying cutting-edge modern research. Starting with the mathematically elegant covariant formalism of the Maxwell equations, from which special relativity derives, the unit covers topics such as the origin of radiation from relativistic particles and from atoms, which are important in astrophysics and particle physics as well as optical and quantum physics. This then introduces the theme of light-matter interactions, which reveals how light can be manipulated and controlled, leading to fascinating phenomena such as optical tweezers, topological insulators and metamaterials. The unique properties and applications of confined electromagnetic waves and their nonlinear interactions are studied in depth, followed by the physics of laser light. The unit is completed with the contemporary research topic of quantum optics. In studying these topics, you will learn advanced theoretical concepts and associated mathematical methods in physics, including tensor calculus, Greens function method, multipole expansion in field theory, and coupled mode theory. By doing this unit, you will be able to synthesise your knowledge of physics and gain new insights into how to identify and apply relevant aspects of physics-based concepts and techniques to solve modern research problems.

Modern astrophysics covers a vast range of scales, from processes within the Solar System which allow for direct testing, to processes that take place in distant places and times, such as the formation of galaxies. Nonetheless, the same physics underpins all of these situations: the plasma of the solar system meets the interstellar medium, which provides the building blocks for galaxies. This unit provides an advanced-level treatment of three major topics in astrophysics: the formation and evolution of galaxies, the structure and morphology of galaxies, and the physics of plasma in our Solar System. You will learn about the behaviour of gas and plasma throughout the Universe, and their effect on phenomena from galaxy structure to space weather. By doing this unit, you will learn how to synthesise your knowledge of physical concepts and processes, and how these concepts and techniques are used to solve modern research problems.

Einstein's General Theory of Relativity represents a pinnacle of modern physics, providing the most accurate description of the action of gravity across the cosmos. To Newton, gravity was simply a force between masses, but Einstein's mathematical language describes gravity in terms of the bending and stretching of space-time. In this course, students will review Einstein's principle of relativity, and the mathematical form of special relativity, and the flat space-time this implies. This will be expanded and generalised to consider Einstein's principle of equivalence and the implications for particle and photon motion with curved space-time. Students will explore the observational consequences of general relativity in several space-time metrics, in particular the Schwarzschild black hole, the Morris-Thorne wormhole, and the Alcubierre warp drive, elucidating the nature of the observer in determining physical quantities. Building on this knowledge, students will understand Einstein's motivation in determining the field equations, relating the distribution of mass and energy to the properties of space-time. Students will apply the field equations, including deriving the cosmological Friedmann-Robertson-Walker metric from the assumption of constant curvature, and using this to determine the universal expansion history and key observables. Students will obtain a complete picture of our modern cosmological model, understanding the constituents of the universe, the need for inflation in the earliest epochs, and the ultimate fate of the cosmos.

Our current understanding of the basic building blocks of matter and interactions between them is called the Standard Model of Particle Physics (SM). This most fundamental description of Nature incorporates three of the four basic interactions which govern how the Universe works, the electromagnetic, weak and strong interactions. This unit investigates the mathematical underpinnings of the SM, a quantum field theory constructed upon fundamental notions of symmetry including Lorentz and local gauge invariance. It also explores the notion of spontaneous symmetry breaking, the Higgs field and the way that fundamental particles acquire mass. The interplay between theory and experiment, which has driven the SM's development, is highlighted. Finally, limitations of the model and possible extensions which could overcome them are discussed. You will learn how the SM is constructed based on symmetry principles, quantum fields and their space-time derivatives; how to derive equations of motion for the fields using the Action Principle; and how predictions for physical observables such as cross sections and decay rates can be calculated starting from the SM Lagrangian density. By studying examples of both recent and historically significant measurements confirming or challenging the SM, you will gain experience in reading and interpreting the scientific literature. Through this unit you will develop an appreciation of humankind's most contemporary and successful attempt to describe Nature in terms of fundamental laws.

Quantum Field Theory (QFT) is the basic mathematical framework that is used for a consistent quantum-mechanical description of relativistic systems, such as fundamental subatomic particles in particle physics. The tools of QFT are also used for description of quasi-particles and critical phenomena in condensed matter physics and other related fields. This course introduces major concepts and technical tools of QFT. The course is largely self-contained and covers also Lagrangian and Hamiltonian formalisms for classical fields, elements of group theory and path integral formulation of quantum mechanics. The main topics include second quantization of various fields and description of their interactions, with the main focus on the most accurate fundamental theory of quantum electromagnetism. The last part of the course deals the concept of the renormalisation group, and its applications to critical phenomena in condensed matter systems. By completing this course, you will obtain knowledge of major concepts and tools of contemporary fundamental physics, that can be employed in a wide range of physics and physics-based research, starting from the description of profound effects in condensed matter physics and ending by the understanding of basic building blocks of the Universe .

Modern nanofabrication and characterisation techniques now allow us to build devices that exhibit controllable quantum features and phenomena. We can now demonstrate the thought experiments posed by the founders of quantum mechanics a century ago, as well as explore the newest breakthroughs in quantum theory. We can also develop new quantum technologies, such as quantum computers. This unit will investigate the latest research results in quantum nanoscience across a variety of platforms. You will be introduced to the latest research papers in the field, published in high-impact journals, and gain an appreciation and understanding of the diverse scientific elements that come together in this research area, including materials, nanofabrication, characterisation, and fundamental theory. You will learn to assess an experiment's demonstration of phenomena in quantum nanoscience, such as quantum coherence and entanglement, mesoscopic transport, exotic topological properties, etc. You will acquire the ability to approach a modern research paper in physics, and to critically analyse the results in the context of the wider scientific community. By doing this unit you will develop the capacity to undertake research, experimental and/or theoretical, in quantum nanoscience.

What is the neural code? How do neural circuits communicate information? What happens in our brain when we make a decision? Computational modelling and theoretical analysis are important tools for addressing these fundamental questions and for determining the functioning mechanisms of the brain. This interdisciplinary unit will provide a thorough and up-to-date introduction to the fields of computational neuroscience and neurophysics. You will learn to develop basic models of how neurons process information and perform quantitative analyses of real neural circuits in action. These models include neural activity dynamics at many different scales, including the biophysical, the circuit and the system levels. Basic data analytics of neural recordings at these levels will also be explored. In addition, you will become familiar with the computational principles underlying perception and cognition, and algorithms of neural adaptation and learning, which will provide knowledge for building-inspired artificial intelligence. Your theoretical learning will be complemented by inquiry-led practical classes that reinforce the above concepts. By doing this unit, you will develop essential modelling and quantitative analysis skills for studying how the brain works.

The need to make sense of confusing, incomplete and noisy data is a problem central to virtually all branches of science. The underlying requirement is to draw robust, unbiased and insightful inferences from the data.After taking this course you should have a working knowledge of common data inference and model-fitting methods, and of machine learning techniques. You should be able to implement the model-fitting algorithms discussed here in your own code and use it to determine parameters from incomplete or noisy data. You will have a conceptual understanding of modern machine-learning techniques, including basic neural networks, and be able to implement your own network to solve a problem. Moreover, you will have the prerequisite knowledge to implement more complex machine learning architectures such as deep learning, using the wide range of available tools. The course is aimed to equip physicists (and other scientists) with practical tools to be deployed in their work, rather than delivering more theoretical content.

Physicists are actively applying their knowledge and technical skills in response to the major challenges of our time, such as environmental change brought about by our soaring demand for energy from finite resources. Their efforts can be found in everyday technology, such as smartphones and GPS devices, which would not exist today without physics. Physics, which underlies the whole of science and technology, has changed our life and society and will change our life and society in the future. This unit will connect the fundamental physics to the modern and future technologies in information and communication technology, energy and sustainability, medicine and health. The information and communication technology module will cover quantum computing, quantum communication, spintronics, optical technology. The energy and sustainability module will cover energy harvesting, energy conversion, energy storage, renewable energy and carbon capture. The medicine and health module will cover nuclear magnetic resonance, medical Imaging, radiation and dosimetry. You will learn the physical concepts and apply these concepts to understanding and evaluating modern technologies in information and communication technology, energy and sustainability, medicine and health. You will also how to track the technologies in principle, and how to predict their impact, and on what time scale. By doing this unit you will develop a deep understanding of these modern and future technologies and prepare yourselves for the related employment.

Condensed matter physics is the science behind semiconductors and all modern electronics, while particle physics describes the very fabric of our Universe. Surprisingly these two seemingly separate aspects of physics use in part very similar formalisms. This unit provides an advanced introduction to both these fields, sharing some coursework with PHYS3936, but going into much more depth through a literature review project offering a critical view of a current research topic in condensed matter physics or particle physics. The particle physics part will introduce the basic constituents of matter, such as quarks and leptons, examining their fundamental properties and interactions. You will gain understanding of extensions to the currently accepted Standard Model of particle physics, and on the relationships between high energy particle physics, cosmology and the early Universe. The condensed matter part will cover the physics that underlies the electromagnetic, thermal, and optical properties of solids. Lectures will include discoveries and new developments in semiconductors, nanostructures, magnetism, and superconductivity, topics which will also be explored in computer lab tutorials. In addition, you will carry out an in-depth critical analysis on a topic of your choice in condensed matter physics and/or particle physics through a literature review research project. In completing this unit, you will gain understanding of the foundations of modern physics and develop research and critical thinking skills.

Looking at the sky it is easy to forget our Sun and the stars are continuous giant nuclear explosions, or that nebulas are vast fields of ionized gases, all obeying the same laws of physics as anything else in the universe. Astrophysics gives us great insight in the larger structures of the universe, and plasma physics is key to understanding matter in space, but also in fusion reactors or for advanced material processing. This unit will provide an advanced introduction to astrophysics and plasma physics, complemented by an extensive literature research project critically investigating a current research topic in plasma physics or astrophysics. You will study three key concepts in astrophysics: the physics of radiation processes, stellar evolution, and binary stars. You will gain understanding of the physics of fundamental phenomena in plasmas. Examples will be given, where appropriate, of the application of these concepts to naturally occurring and man-made plasmas. In addition, you will carry out an in-depth critical analysis of a topic or your choice in astrophysics and/or plasma physics through a literature review research project. In completing this unit you will gain understanding of the foundations of modern physics and develop research and critical thinking skills.

"If you can't explain it simply, you don't understand it well enough". This quote is widely attributed to Albert Einstein, but regardless of its provenance, it suggests that one measure of an expert's knowledge can be found in their ability to translate complex ideas so that they are accessible to anyone. The communication of science to the public is essential for science and society. In order to increase public understanding and appreciation of science, researchers must be able to explain their results, and the wider context of their research, to non-experts. This unit will explore some theoretical foundations of science communications, identify outstanding practitioners and empower students to produce effective science communication in different media. In this unit you will learn the necessary skills and techniques to tell engaging and informative science stories in order to bring complex ideas to life, for non-expert audiences. By undertaking this unit you will develop a greater understanding of the wider context of your honours unit, advance your communication skills and be able to explain your honours research to non-expert audiences such as friends, family or future employers. These transferable skills will equip you for future research - where emphasis is increasingly placed on public communication and/or outreach - or professional pathways - where effective communication of complex ideas is highly valued.

An indispensable attribute of an effective scientific researcher is the ability to collect, analyse and interpret data. Central to this process is the ability to create hypotheses and test these by using rigorous experimental designs. This modular unit of study will introduce the key concepts of experimental design and data analysis. Specifically, you will learn to formulate experimental aims to test a specific hypothesis. You will develop the skills and understanding required to design a rigorous scientific experiment, including an understanding of concepts such as controls, replicates, sample size, dependent and independent variables and good research practice (e.g. blinding, randomisation). By completing this unit you will develop the knowledge and skills required to appropriately analyse and interpret data in order to draw conclusions in the context of an advanced research project. From this unit of study, you will emerge with a comprehensive understanding of how to optimise the design and analysis of an experiment to most effectively answer scientific questions.

In the contemporary world, a wide variety of ethical concerns impinge upon the practice of scientific research. In this unit, you will learn how to identify potential ethical issues within science, acquire the tools necessary to analyse them, and develop the ability to articulate ethically sound insights about how to resolve them. In the first portion of the unit, you will be familiarised with how significant developments in post-World War II science motivated sustained ethical debate among scientists and in society. In the second portion of the unit, you will select from either a Human Ethics module or an Animal Ethics module and learn the requirements of how to ensure your research complies with appropriate national legislation and codes of conduct. By undertaking this unit, you will develop the ability to conduct scientific research in an ethically justifiable way, place scientific developments and their application in a broader social context, and analyse the social implications and ethical issues that may potentially arise in the course of developing scientific knowledge.

This unit of study explores a variety of issues in philosophy of science. It uses a range of examples from different sciences (e.g., physics, biology, chemistry) to examine theories, models, concepts, and the various tools that are considered part of the scientific method. Students will learn how to apply this philosophical analysis of science to historical and contemporary findings, as well as to their own research.

Topology, developed at the end of the 19th Century to investigate the subtle interaction of analysis and geometry, is now one of the basic disciplines of mathematics. A working knowledge of the language and concepts of topology is essential in fields as diverse as algebraic number theory and non-linear analysis. This unit develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory. Topics covered include: Metric spaces, convergence, completeness and the Contraction Mapping Theorem; Metric topology, open and closed subsets; Topological spaces, subspaces, product spaces; Continuous mappings and homeomorphisms; Compactness Connectedness Hausdorff spaces and normal spaces. You will learn methods and techniques of proving basic theorems in point-set topology and apply them to other areas of mathematics including basic Hilbert space theory and abstract Fourier series. By doing this unit you will develop solid foundations in the more formal aspects of topology, including knowledge of abstract concepts and how to apply them. Applications include the use of the Contraction Mapping Theorem to solve integral and differential equations.

This unit of study lies at the heart of modern algebra. In the unit we investigate the mathematical theory that was originally developed for the purpose of studying polynomial equations. In a nutshell, the philosophy is that it should be possible to completely factorise any polynomial into a product of linear factors by working over a large enough field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalises the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions. Along the way you will see some beautiful gems of mathematics, including Fermat's Theorem on primes expressible as a sum of two squares, solutions to the ancient Greek problems of trisecting the angle, squaring the circle, and doubling the cube, and the crown of the course: Galois' proof that there is no analogue of the quadratic formula for the general quintic equation. On completing this unit of study you will have obtained a deep understanding of modern abstract algebra.

The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods. The theory has many applications and stimulates new developments in almost all areas of mathematics. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, linearisation, and analysis of asymptotic behaviour. The applications in this unit will be drawn from predator-prey systems, population models, chemical reactions, and other equations and systems from mathematical biology. You will learn how to use ordinary differential equations to model biological, chemical, physical and/or economic systems and how to use different methods from dynamical systems theory and the theory of nonlinear ordinary differential equations to find the qualitative outcome of the models. By doing this unit you will develop skills in using and analysing nonlinear differential equations which will prepare you for further studies in mathematics, systems biology or physics or for careers in mathematical modelling.

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. For students, this provides the first taste of the investigation on the deep relation between geometry and topology of mathematical objects, highlighted in the classic Gauss-Bonnet Theorem. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to remind the students about all the content covered in the mathematical units for previous years, most importantly the key ideas in vector calculus, along with some applications. It also helps to prepare the students for honours courses like Riemannian Geometry. By doing this unit you will further appreciate the beauty of mathematics which originated from the need to solve practical problems, develop skills in understanding the geometry of the surrounding environment, prepare yourself for future study or the workplace by developing advanced critical thinking skills and gain a deep understanding of the underlying rules of the Universe.

Measure theory is the study of fundamental ideas as length, area, volume, arc length and surface area. It is the basis for Lebesgue integration theory used in advanced mathematics ever since its development in about 1900. Measure theory is also a key foundation for modern probability theory. The course starts by establishing the basics of measure theory and the theory of Lebesgue integration, including important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. These ideas are applied to Fourier Analysis which leads to results such as the Inversion Formula and Plancherel's Theorem. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Key ideas of this theory are applied in detail to probability theory to provide a rigorous framework for probability which takes in and generalizes familiar ideas such as distributions and conditional expectation. When you complete this unit you will have acquired a new generalized way of thinking about key mathematical concepts such as length, area, integration and probability. This will give you a powerful set of intellectual tools and equip you for further study in mathematics and probability.

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimisation problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straightforward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

Fluid Dynamics is the study of systems which allow for a macroscopic description in some continuum limit. It is not limited to the study of liquids such as water but includes our atmosphere and even car traffic. Whether a system can be treated as a fluid, depends on the spatial scales involved. Fluid dynamics presents a cornerstone of applied mathematics and comprises a whole gamut of different mathematical techniques, depending on the question we ask of the system under consideration. The course will discuss applications from engineering, physics and mathematics: How and in what situations a system which is not necessarily liquid can be described as a fluid? The link between an Eulerian description of a fluid and a Lagrangian description of a fluid, the basic variables used to describe flows, the need for continuity, momentum and energy equations, simple forms of these equations, geometric and physical simplifying assumptions, streamlines and stream functions, incompressibility and irrotationality and simple examples of irrotational flows. By the end of this unit, students will have received a basic understanding into fluid mechanics and have acquired general methodology which they can apply in their further studies in mathematics and/or in their chosen discipline.

Sophisticated mathematics and numerical programming underlie many computer applications, including weather forecasting, computer security, video games, and computer aided design. This unit of study provides a strong foundational introduction to modern interactive programming, computational algorithms, and numerical analysis. Topics covered include: (I) basics ingredients of programming languages such as syntax, data structures, control structures, memory management and visualisation; (II) basic algorithmic concepts including binary and decimal representations, iteration, linear operations, sources of error, divide-and-concur, algorithmic complexity; and (III) basic numerical schemes for rootfinding, integration/differentiation, differential equations, fast Fourier transforms, Monte Carlo methods, data fitting, discrete and continuous optimisation. You will also learn about the philosophical underpinning of computational mathematics including the emergence of complex behaviour from simple rules, undecidability, modelling the physical world, and the joys of experimental mathematics. When you complete this unit you will have a clear and comprehensive understanding of the building blocks of modern computational methods and the ability to start combining them together in different ways. Mathematics and computing are like cooking. Fundamentally, all you have is sugar, fat, salt, heat, stirring, chopping. But becoming a good chef requires knowing just how to put things together in creative ways that work. In a previous study, you should have learned to cook. Now you're going to learn how to make something someone else might want to pay for more than one time.

Lagrangian and Hamiltonian dynamics are reformulations of classical Newtonian mechanics into a mathematically sophisticated framework using arbitrary coordinate systems. This formulation of classical mechanics generalises elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamics from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to simplify apparently complicated dynamical problems. Connections between geometry and different physical theories beyond classical mechanics are explored. Students will be expected to describe and solve mechanical systems of some complexity including planetary motion and to investigate stability. Hamilton-Jacobi theory will be used to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. Students will study an application of Lagrangian and Hamiltonian dynamics described in a modern research paper.

The aim of this unit is to introduce some fundamental concepts of the theory of partial differential equations (PDEs) arising in Physics, Chemistry, Biology and Mathematical Finance. The focus is mainly on linear equations but some important examples of nonlinear equations and related phenomena re introduced as well. After an introductory lecture, we proceed with first-order PDEs and the method of characteristics. Here, we also nonlinear transport equations and shock waves are discussed. Then the theory of the elliptic equations is presented with an emphasis on eigenvalue problems and their application to solve parabolic and hyperbolic initial boundary-value problems. The Maximum principle and Harnack's inequality will be discussed and the theory of Green's functions.

The unit will begin with a revision of properties of complex numbers and complex functions. This will be followed by material on conformal mappings, Riemann surfaces, complex integration, entire and analytic functions, the Riemann mapping theorem, analytic continuation, and Gamma and Zeta functions. Finally, special topics chosen by the lecturer will be presented, which may include elliptic functions, normal families, Julia sets, functions of several complex variables, or complex manifolds.

One of the most important aims of algebraic topology is to distinguish or classify topological spaces and maps between them up to homeomorphism. Invariants and obstructions are key to achieve this aim. A familiar invariant is the Euler characteristic of a topological space, which was initially discovered via combinatorial methods and has been rediscovered in many different guises. Modern algebraic topology allows the solution of complicated geometric problems with algebraic methods. Imagine a closed loop of string that looks knotted in space. How would you tell if you can wiggle it about to form an unknotted loop without cutting the string? The space of all deformations of the loop is an intractable set. The key idea is to associate algebraic structures, such as groups or vector spaces, with topological objects such as knots, in such a way that complicated topological questions can be phrased as simpler questions about the algebraic structures. In particular, this turns questions about an intractable set into a conceptual or finite, computational framework that allows us to answer these questions with certainty. In this unit you will learn about fundamental group and covering spaces, homology and cohomology theory. These form the basis for applications in other domains within mathematics and other disciplines, such as physics or biology. At the end of this unit you will have a broad and coherent knowledge of Algebraic Topology, and you will have developed the skills to determine whether seemingly intractable problems can be solved with topological methods.

Commutative Algebra provides the foundation to study modern uses of Algebra in a wide array of settings, from within Mathematics and beyond. The techniques of Commutative Algebra underpin some of the most important advances of mathematics in the last century, most notably in Algebraic Geometry and Algebraic Topology. This unit will teach students the core ideas, theorems, and techniques from Commutative Algebra, and provide examples of their basic applications. Topics covered include affine varieties, Noetherian rings, Hilbert basis theorem, localisation, the Nullstellansatz, ring specta, homological algebra, and dimension theory. Applications may include topics in scheme theory, intersection theory, and algebraic number theory. On completion of this unit students will be thoroughly prepared to undertake further study in algebraic geometry, algebraic number theory, and other areas of mathematics. Students will also gain facility with important examples of abstract ideas with far-reaching consequences.

Functional analysis is one of the major areas of modern mathematics. It can be thought of as an infinite-dimensional generalisation of linear algebra and involves the study of various properties of linear continuous transformations on normed infinite-dimensional spaces. Functional analysis plays a fundamental role in the theory of differential equations, particularly partial differential equations, representation theory, and probability. In this unit you will cover topics that include normed vector spaces, completions and Banach spaces; linear operators and operator norms; Hilbert spaces and the Stone-Weierstrass theorem; uniform boundedness and the open mapping theorem; dual spaces and the Hahn-Banach theorem; and spectral theory of compact self-adjoint operators. A thorough mechanistic grounding in these topics will lead to the development of your compositional skills in the formulation of solutions to multifaceted problems. By completing this unit you will become proficient in using a set of standard tools that are foundational in modern mathematics and will be equipped to proceed to research projects in PDEs, applied dynamics, representation theory, probability, and ergodic theory.

Representation theory is the abstract study of the possible types of symmetry in all dimensions. It is a fundamental area of algebra with applications throughout mathematics and physics: the methods of representation theory lead to conceptual and practical simplification of any problem in linear algebra where symmetry is present. This unit will introduce you to the basic notions of modules over associative algebras and representations of groups, and the ways in which these objects can be classified. You will learn the special properties that distinguish the representation theory of finite groups over the complex numbers, and also the unifying principles which are common to the representation theory of a wider range of algebraic structures. By learning the key concepts of representation theory you will also start to appreciate the power of category-theoretic approaches to mathematics. The mental framework you will acquire from this unit of study will enable you both to solve computational problems in linear algebra and to create new mathematical theory.

Variational and spectral methods are foundational in mathematical models that govern the configurations of many physical systems. They have wide-ranging applications in areas such as physics, engineering, economics, differential geometry, optimal control and numerical analysis. In addition they provide the framework for many important questions in modern geometric analysis. This unit will introduce you to a suite of methods and techniques that have been developed to handle these problems. You will learn the important theoretical advances, along with their applications to areas of contemporary research. Special emphasis will be placed on Sobolev spaces and their embedding theorems, which lie at the heart of the modern theory of partial differential equations. Besides engaging with functional analytic methods such as energy methods on Hilbert spaces, you will also develop a broad knowledge of other variational and spectral approaches. These will be selected from areas such as phase space methods, minimax theorems, the Mountain Pass theorem or other tools in the critical point theory. This unit will equip you with a powerful arsenal of methods applicable to many linear and nonlinear problems, setting a strong foundation for understanding the equilibrium or steady state solutions for fundamental models of applied mathematics.

Computational mathematics fulfils two distinct purposes within Mathematics. On the one hand the computer is a mathematician's laboratory in which to model problems too hard for analytical treatment and to test existing theories; on the other hand, computational needs both require and inspire the development of new mathematics. Computational methods are an essential part of the tool box of any mathematician. This unit will introduce you to a suite of computational methods and highlight the fruitful interplay between analytical understanding and computational practice. In particular, you will learn both the theory and use of numerical methods to simulate partial differential equations, how numerical schemes determine the stability of your method and how to assure stability when simulating Hamiltonian systems, how to simulate stochastic differential equations, as well as modern approaches to distilling relevant information from data using machine learning. By doing this unit you will develop a broad knowledge of advanced methods and techniques in computational applied mathematics and know how to use these in practice. This will provide a strong foundation for research or further study.

Mathematical approaches to many real-world problems are underpinned by powerful and wide ranging mathematical methods and techniques that have become standard in the field and should be in the toolbag of all applied mathematicians. This unit will introduce you to a suite of those methods and give you the opportunity to engage with applications of these methods to well-known problems. In particular, you will learn both the theory and use of asymptotic methods which are ubiquitous in applications requiring differential equations or other continuous models. You will also engage with methods for probabilistic models including information theory and stochastic models. By doing this unit you will develop a broad knowledge of advanced methods and techniques in applied mathematics and know how to use these in practice. This will provide a strong foundation for using mathematics in a broad sweep of practical applications in research, in industry or in further study.

Applied Mathematics harnesses the power of mathematics to give insight into phenomena in the wider world and to solve practical problems. Modelling is the key process that translates a scientific or other phenomenon into a mathematical framework through applying suitable assumptions, identifying important variables and deriving a well-defined mathematical problem. Mathematicians then use this model to explore the real-world phenomenon, including making predictions. Good mathematical modelling is something of an art and is best learnt by example and by writing, refining and analysing your own models. This unit will introduce you to some classic mathematical models and give you the opportunity to analyse, explore and extend these models to make predictions and gain insights into the underlying phenomena. You will also engage with modelling in depth in at least one area of application. By doing this unit you will develop a broad knowledge of advanced mathematical modelling methods and techniques and know how to use these in practice. This will provide a strong foundation for applying mathematics and modelling to many diverse applications and for research or further study.

In applied mathematics, dynamical systems are systems whose state is changing with time. Examples include the motion of a pendulum, the change in the population of insects in a field or fluid flow in a river. These systems are typically represented mathematically by differential equations or difference equations. Dynamical systems theory reveals universal mechanisms behind disparate natural phenomena. This area of mathematics brings together sophisticated theory from many areas of pure and applied mathematics to create powerful methods that are used to understand and control the dynamical building blocks which make up physical, biological, chemical, engineered and even sociological systems. By doing this unit you will develop a broad knowledge of methods and techniques in dynamical systems, and know how to use these to analyse systems in nature and in technology. This will provide a strong foundation for using mathematics in a broad sweep of applications and for research or further study.

The aim of Financial Mathematics is to establish a theoretical background for building models of securities markets and provides computational techniques for pricing financial derivatives and risk assessment and mitigation. Specialists in Financial Mathematics are widely sought after by major investment banks, hedge funds and other, government and private, financial institutions worldwide. This course is foundational for honours and masters programs in Financial Mathematics. Its aim is to introduce the basic concepts and problems of securities markets and to develop theoretical frameworks and computational tools for pricing financial products and hedging the risk associated with them. This unit will focus on two ideas that are fundamental for Financial Mathematics. You will learn how the concept of arbitrage and the concept of martingale measure provide a unified approach to a large variety of seemingly unrelated problems arising in practice. You will also learn how to use the wide range of tools required by Financial Mathematics, including stochastic calculus, partial differential equations, optimisation and statistics. By doing this unit, you will learn how to formulate problems that arise in finance as mathematical problems and how to solve them using the concepts of arbitrage and martingale measure. You will also learn how to choose an appropriate computational method and devise explicit numerical algorithms useful for a practitioner.

Capturing random phenomena is a challenging problem in many disciplines from biology, chemistry and physics through engineering to economics and finance. There is a wide spectrum of problems in these fields, which are described using random processes that evolve with time. Hence it is of crucial importance that applied mathematicians are equipped with tools used to analyse and quantify random phenomena. This unit will introduce an important class of stochastic processes, using the theory of martingales. You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. Special attention will be given to the classical notion of the Brownian motion, which is the most celebrated and widely used example of a continuous martingale. By completing this unit, you will learn how to rigorously describe and tackle the evolution of random phenomena using continuous time stochastic processes. You will also gain a deep knowledge about stochastic integration, which is an indispensable tool to study problems arising, for example, in Financial Mathematics.

Securities and derivatives are the foundation of modern financial markets. The fixed-income market, for example, is the dominant sector of the global financial market where various interest-rate linked securities are traded, such as zero-coupon and coupon bonds, interest rate swaps and swaptions. This unit will investigate short-term interest rate models, the Heath-Jarrow-Morton approach to instantaneous forward rates and recently developed models of forward London Interbank Offered Rates (LIBORs) and forward swap rates. You will learn about pricing and hedging of credit derivatives, another challenging and practically important problem and become familiar with stochastic models for credit events, dependent default times and credit ratings. You will learn how to value and hedge single-name and multi-name credit derivatives such as vulnerable options, corporate bonds, credit default swaps and collateralized debt obligations. You will also learn about the most recent developments in Financial Mathematics, such as robust pricing and nonlinear evaluations. By doing this unit, you will get a solid grasp of mathematical tools used in valuation and hedging of fixed income securities, develop a broad knowledge of advanced quantitative methods related to interest rates and credit risk and you will learn to use powerful mathematical tools to address important real-world quantitative problems in the finance industry.

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. In this unit you will rigorously establish the basic properties and limit theory of discrete-time Markov chains and branching processes and then, building on this foundation, derive key results for the Poisson process and continuous-time Markov chains, stopping times and martingales. You will learn about various illustrative examples throughout the unit to demonstrate how stochastic processes can be applied in modelling and analysing problems of practical interest, such as queuing, inventory, population, financial asset price dynamics and image processing. By completing this unit, you will develop a solid mathematical foundation in stochastic processes which will become the platform for further studies in advanced areas such as stochastic analysis, stochastic differential equations, stochastic control and financial mathematics.

Classical linear models are widely used in science, business, economics and technology. This unit will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using linear methods, together with concepts of collection of data and design of experiments. You will first consider linear models and regression methods with diagnostics for checking appropriateness of models, looking briefly at robust regression methods. Then you will consider the design and analysis of experiments considering notions of replication, randomisation and ideas of factorial designs. Throughout the course you will use the R statistical package to give analyses and graphical displays. This unit includes material in STAT3022 Applied Linear Models, but has an additional component on the mathematical techniques underlying applied linear models together with proofs of distribution theory based on vector space methods.

In today's data-rich world, more and more people from diverse fields need to perform statistical analyses, and indeed there are more and more tools to do this becoming available. It is relatively easy to "point and click" and obtain some statistical analysis of your data. But how do you know if any particular analysis is indeed appropriate? Is there another procedure or workflow which would be more suitable? Is there such a thing as a "best possible" approach in a given situation? All of these questions (and more) are addressed in this unit. You will study the foundational core of modern statistical inference, including classical and cutting-edge theory and methods of mathematical statistics with a particular focus on various notions of optimality. The first part of the unit covers aspects of distribution theory which are applied in the second part which deals with optimal procedures in estimation and testing. The framework of statistical decision theory is used to unify many of the concepts that are introduced in this unit. You will rigorously prove key results and apply these to real-world problems in laboratory sessions. By completing this unit, you will develop the necessary skills to confidently choose the best statistical analysis to use in many situations.

In our ever-changing world, we are facing a new data-driven era where the capability to efficiently combine and analyse large data collections is essential for informed decision making in business and government, and for scientific research. Statistics and data analytics consulting provide an important framework for many individuals to seek assistance with statistics and data-driven problems. This unit of study will provide students with an opportunity to gain real-life experience in statistical consulting or work with collaborative (interdisciplinary) research. In this unit, you will have an opportunity to have practical experience in a consultation setting with real clients. You will also apply your statistical knowledge in a diverse collection of consulting projects while learning project and time management skills. In this unit you will need to identify and place the client's problem into an analytical framework, provide a solution within a given time frame and communicate your findings back to the client. All such skills are highly valued by employers. This unit will foster the expertise needed to work in a statistical consulting firm or data analytical team which will be essential for data-driven professional and research pathways in the future.

Applied Statistics fundamentally brings statistical learning to the wider world. Some data sets are complex due to the nature of their responses or predictors or have high dimensionality. These types of data pose theoretical, methodological and computational challenges that require knowledge of advanced modelling techniques, estimation methodologies and model selection skills. In this unit you will investigate contemporary model building, estimation and selection approaches for linear and generalised linear regression models. You will learn about two scenarios in model building: when an extensive search of the model space is possible; and when the dimension is large and either stepwise algorithms or regularisation techniques have to be employed to identify good models. These particular data analysis skills have been foundational in developing modern ideas about science, medicine, economics and society and in the development of new technology and should be in the toolkit of all applied statisticians. This unit will provide you with a strong foundation of critical thinking about statistical modelling and technology and give you the opportunity to engage with applications of these methods across a wide scope of applications and for research or further study.

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory and applies it to problems in mathematical statistics. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that bring together both the theory of discrete random variables and the theory of continuous random variables that were introduce to earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and also verify important results in Mathematical Statistics. These involve exponential families, efficient estimation, large-sample testing and Bayesian methods. Finally you will verify important convergence properties of the expectation-maximisation (EM) algorithm. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability or mathematical statistics.

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory (based on measure theory) that was developed by Andrey Kolmogorov. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that brings together both the theory of discrete random variables and the theory of continuous random variables that were introduced earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, conditional expectation, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and to thoroughly study discrete-time martingales. Originally used to model betting strategies, martingales are a powerful generalisation of random walks that allow us to prove fundamental results such as the Strong Law of Large Numbers or analyse problems such as the gambler's ruin. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability, stochastic systems or financial mathematics.

Research is the lifeblood of the physical sciences, and participation in research is both a privilege and an opportunity for learning and growth. In this unit you will complete a research project in physics. Together with your supervisor, you will identify a novel research question and a method to address it. Typical activities during the project include review and synthesis of the literature, specialised study in a sub-field of physics, design and conduct of experiments, data mining, theoretical work, modelling, calculation, simulation, coding, and the use of specialised software. Your project will be assessed through an oral presentation to your peers and colleagues, and a forty-page research report. Successful completion of your honours research project will demonstrate that you have mastered significant research and professional skills for undertaking postgraduate studies, or a range of careers in industry.

Research is the lifeblood of the physical sciences, and participation in research is both a privilege and an opportunity for learning and growth. In this unit you will complete a research project in physics. Together with your supervisor, you will identify a novel research question and a method to address it. Typical activities during the project include review and synthesis of the literature, specialised study in a sub-field of physics, design and conduct of experiments, data mining, theoretical work, modelling, calculation, simulation, coding, and the use of specialised software. Your project will be assessed through an oral presentation to your peers and colleagues, and a forty-page research report. Successful completion of your honours research project will demonstrate that you have mastered significant research and professional skills for undertaking postgraduate studies, or a range of careers in industry.

Research is the lifeblood of the physical sciences, and participation in research is both a privilege and an opportunity for learning and growth. In this unit you will complete a research project in physics. Together with your supervisor, you will identify a novel research question and a method to address it. Typical activities during the project include review and synthesis of the literature, specialised study in a sub-field of physics, design and conduct of experiments, data mining, theoretical work, modelling, calculation, simulation, coding, and the use of specialised software. Your project will be assessed through an oral presentation to your peers and colleagues, and a forty-page research report. Successful completion of your honours research project will demonstrate that you have mastered significant research and professional skills for undertaking postgraduate studies, or a range of careers in industry.

Research is the lifeblood of the physical sciences, and participation in research is both a privilege and an opportunity for learning and growth. In this unit you will complete a research project in physics. Together with your supervisor, you will identify a novel research question and a method to address it. Typical activities during the project include review and synthesis of the literature, specialised study in a sub-field of physics, design and conduct of experiments, data mining, theoretical work, modelling, calculation, simulation, coding, and the use of specialised software. Your project will be assessed through an oral presentation to your peers and colleagues, and a forty-page research report. Successful completion of your honours research project will demonstrate that you have mastered significant research and professional skills for undertaking postgraduate studies, or a range of careers in industry.

All students in Science Honours must enrol in this non-assessable unit of study in their final semester. This unit will contain your final Honours mark as calculated from your coursework and research project units.